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\title{复变函数第1章 - 复数}
\author{CG ET AL}
%\date{2025年9月6日}

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\begin{document}

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% 封面页
\begin{frame}
  \titlepage
\end{frame}

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% 目录页
\begin{frame}{目录}
  \begin{multicols}{2}
  \tableofcontents
  \end{multicols}
\end{frame}

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%1.1
\section{算术运算}
\begin{frame}{1.1. 算术运算}
定义复数的加减乘除运算。

\end{frame}

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%1.2
\section{平方根}
\begin{frame}{1.2. 平方根}

计算复数 $\alpha+i\beta$ 的平方根。

\end{frame}

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%1.3
\section{合理性}
\begin{frame}{1.3. 合理性}

什么是复数域？
    
设 $F$ 是包含实数域，且使得 $x^2+1=0$ 有解的最小的域。
证明 $\mathbb{C}$ 是 $F$ 的一个子域。

证明存在域的同构 $\mathbb{C}$ 与 $\mathbb{R}[x]/(x^2+1)$. 


\end{frame}

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%1.4
\section{共轭与绝对值}
\begin{frame}{1.4. 共轭与绝对值}
  
定义复数的共轭与绝对值。

证明复数的拉格朗日恒等式。

\end{frame}

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%1.5
\section{不等式}
\begin{frame}{1.5. 不等式}
  
证明复数的三角不等式。

证明复数的柯西不等式。

\end{frame}

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%1.6
\section{复数的几何表示}
\begin{frame}{1.6. 复数的几何表示}
  
什么是复平面？

使用复平面上的几何图形来计算复数的加减乘除运算。

将一个复数理解为二维实向量空间$\mathbb{R}^2$上的线性变换。

用矩阵的乘法来实现复数的乘法。


\end{frame}

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%1.7
\section{高次单位根}
\begin{frame}{1.7. 高次单位根}
  
求复数的$n$次单位根。

\end{frame}

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%1.8
\section{解析几何}
\begin{frame}{1.8. 解析几何}
  
用复数的方程来表示平面上的点、直线、圆、椭圆、双曲线、抛物线。

\end{frame}

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%1.9
\section{球面表示}
\begin{frame}{1.9. 球面表示}
 
什么是黎曼球面？

什么是球极投影？

证明球极投影建立了球面上的圆周与平面上的圆周和直线之间的对应。

通过球极投影定义复平面上的球面度量。


\end{frame}

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%1.10
\section{习题一}
\begin{frame}[allowframebreaks]{习题一 }

\begin{enumerate}

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\item  %1

计算下列各数:
 $\sqrt{i}$\quad  
 $\sqrt{-i}$\quad 
 $\sqrt{1+i}$\quad 
 $\sqrt[4]{-1}$\quad 
 $\sqrt[6]{i}$\quad 
 $\sqrt[6]{-i}$\quad 
 $\sqrt{\frac{1-i\sqrt{3}}{2}}$
    
\newpage 
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\item  %2

解二次方程 $x^2 + (\alpha + i\beta)x + \gamma + i\delta = 0$.


\newpage 
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\item  %3

计算 $$\frac{z}{z^2+1}$$ 其中 $z=x+iy$ 或者 $z=x-iy$, 并验证两个结果共轭。


\newpage 
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\item  %4

求下列复数的模:
 $$-2i(3+i)(2+4i)(1+i)$$
 $$\frac{(3+4i)(-1+2i)}{(-1-i)(3-i)}$$


\newpage 
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\item  %5

当 $|a|=1$ 或者 $|b|=1$, 且 $\bar{a}b \neq 1$ 时, 证明
$$
\left|\frac{a-b}{1-\bar{a}b}\right| = 1.
$$


\newpage 
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\item  %6

当 $|a|<1$ 且 $|b|<1$ 时, 证明
$$
\left|\frac{a-b}{1-\bar{a}b}\right| < 1.
$$
    

\newpage 
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\item  %7
当 $|a_i|<1$, $\lambda_i \geq 0$ ($i=1,2,\cdots,n$) 且 $\lambda_1+\lambda_2+\cdots+\lambda_n=1$ 时, 证明
    $$
    |\lambda_1 a_1 + \lambda_2 a_2 + \cdots + \lambda_n a_n| < 1.
    $$

\newpage 

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\item  %8
    
证明存在复数 $z$ 满足 $|z-a|+|z+a|=2|c|$ 当且仅当 $|a| \leq |c|$. 

如果条件成立, 求 $|z|$ 的最大值和最小值。

\newpage 

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\item  %9
    
求点 $a \in \mathbb{C}$ 关于坐标轴分角线的对称点。

\newpage 

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\item  %10
    
证明点 $a_1, a_2, a_3$ 为等边三角形的三个顶点当且仅当 

$$
a_1^2 + a_2^2 + a_3^2 = a_1a_2 + a_2a_3 + a_3a_1.
$$

\newpage 

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\item  %11
    
设 $a$ 和 $b$ 为正方形的两个顶点，求另外两个顶点。

\newpage 

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\item  %12
    
设一个三角形的三个顶点分别为 $a_1, a_2, a_3$, 求外接圆的圆心和半径。


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\item  %13
    
证明三角形的内角和为 $\pi$.

\newpage 

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\item  %14
    
将 $\cos 3\theta$, $\cos 4\theta$ 与 $\sin 5\theta$ 用 $\cos \theta$ 和 $\sin \theta$ 表示。

\newpage 

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\item  %15
    
简化 

$$
1 + \cos \theta + \cos 2\theta + \cdots + \cos n\theta
$$ 

和 

$$
1 + \sin \theta + \sin 2\theta + \cdots + \sin n\theta
$$

其中 $n \geq 2$ 为正整数。

\newpage 

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\item  %16
    
用代数形式表示 5 次单位根和 10 次单位根。

\newpage 

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\item  %17
    
设 
$$
\omega = \cos \frac{2\pi}{n} + i \sin \frac{2\pi}{n}
$$
其中 $n \geq 2$ 为正整数。

(1) 证明对任意不是 $n$ 的整数倍的整数 $k$, 有 
$$
1 + \omega^k + \cdots + \omega^{(n-1)k} = 0
$$

(2) 求值
$$
1 - \omega^k + \omega^{2k} - \cdots + (-1)^{n-1} \omega^{(n-1)k}$$


\newpage 

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\item  %18
    
证明平行四边形的对角线相互平分而菱形的对角线相互正交。

\newpage 

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\item  %19
    
证明一个圆的平行弦的中点在垂直于这些弦的直径上。

\newpage 

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\item  %20

证明过 $a \in \mathbb{C}$ 与 $\frac{1}{\bar{a}}$ 的所有圆周都与单位圆周正交。

\newpage 

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\item  %21
    
分析方程 $az + b\bar{z} + c = 0$ 所表示的几何图形。（$a,b,c$是复数）

\newpage 

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\item  %22
    
证明点 $z$ 和 $z{\,}'$ 位于 Riemann 球面上一条直径的两个端点当且仅当 $zz{\,}' = -1$.

\newpage 

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\item  %23
    
一个立方体的所有顶点都在单位球面上，各棱平行于坐标轴，求各顶点的球极投影的像。

\newpage 

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\item  %24
    
设 $z, z{\,}' \in \mathbb{C}$ 为不同的点，$X, X{\,}'$ 为它们在球极投影下的原像，$N = (0, 0, 1)$. 证明 $\triangle NXX{\,}'$ 与 $\triangle NZZ{\,}'$ 相似。

\newpage 

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\item  %25
    
求圆心为 $a$、半径为 $R$ 的圆周在球极投影下的原像的半径。

\end{enumerate}

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\end{document}

